You don't need to compute the phase, just the levels, then a threshold detector.

The Goertzel_algorithm has frequency "bins" the higher the sampling rate the narrower the freq bins.

Eg frequency bin 7 might cover frequencies from 300Hz to 400Hz so the bin width is 100Hz.. a faster sampling rate might have a bin freq width of 75Hz.

BUT BUT BUT

While the micro is detecting tones.... there's not much time resources left for the micro.... this is something you must consider.

Google Goertzel -- this should give you lots of hits.

Joe